f(k) =
The ltered output is shown in the picture of g. 5.1c and iso-surfaces of the same are seen in the picture of g. 5.1d. The lter cut-o is placed at k ∼ 70. From these two pictures it can be inferred that the magnetic eld has several regions of high concentration. But these regions are like clumps of several small scales and do not show any denite expected large-scale structures. Modeling of the iso-surfaces points to the fact that the inherent features in the eld are fractal in nature, as a result of highk random forcing used here (see section 5.3 for the modeling), i.e. the iso-surfaces are neither one-dimensional nor two-dimensional structures but have a co-dimension of 1.5 (see section 5.3). This ltered system is what is studied in the forced case, for understanding the structures, structure functions and other statistical properties of the forced turbulent system. From here on the reference to forced case implies the output obtained from the cut-o lter. Also all these features are studied for the new strategy (termed here as special case) in which the forcing is stopped, allowing decaying turbulence to take charge, to form the large-scale magnetic structures (the idea explained in detail in section 5.5.3). Decaying case is also reported separately.
5.2 Structure Functions and ESS
The denition of a structure function has already been mentioned in section 3.3 (which was arrived at using equations (3.1) and (3.2)). This equation was written there in terms of velocity. For many other quantities, a similar equation could be written. Thus a general form of the equation is now reproduced here as:
δc` = [c(r+`)−c(r)]·`/` (5.2) c`=
δc`21/2
. (5.3)
Spc(`) =hδc`pi ∼`ζp, (5.4) Hereζp is a constant, p-dependent scaling exponent andcis any quantity like velocity or magnetic eld or Elsässer variable. Note that the equation (5.4) is only valid in the inertial (scaling) range(s) of the spectra.
Here it is also to be noted that only velocity is Galilean invariant and is the primary quantity that is plotted in the 2D-hydrodynamic case. The Elsässer variablez±, repre-sents the total energy which is an ideal invariant in 3D-MHD (not Galilean invariant), under the assumption of negligible cross helicity (i.e. when HC ∼ 0, then (z+)2 ∼ (z−)2 andE ∼ 14(z+)2). Magnetic eld is not an invariant of any form. However, since total energy is dominated by magnetic energy and also since the interest is on under-standing magnetic eld structures; here structure functions for magnetic eld andz+, are plotted.
Structure Functions
In general, from the equation (5.1) several orders of structure functions can be plot-ted. In this work the structure functions of order 1 to 8 are plotplot-ted. Convexity and monotonicity constraints ([1]) are not applicable to odd order structure function scal-ing exponents, so there is a chance that the odd order structure functions can become negative (see [1]). Hence all the structure functions plotted here are calculated from the absolute values of eld increments, avoiding cancellation eects, in the averaging process. Considering the fact that higher order structure functions suer from severe statistical convergence errors, in this work the order of structure functions plotted is limited to eight orders, although it is possible to plot many higher order structure functions. Using the extended self similarity approach (see below) these eight orders of structure functions accurately and adequately represent the scaling behavior in the structures. Figures 5.2a,b and 5.2c,d are the structure function plots of z+ and b re-spectively. Figure 5.2a represents the structure functions at states closer to the initial states of the systems (exact times mentioned in the caption of the gure), for z+ in all the three cases mentioned in section 5.1. Figure 5.2b is the nal state of structure functions ofz+. Figures 5.2c and 5.2d are structure functions ofbat the same instances respectively. On all these graphs the x-axis is from 0 to 2π, the limits of the bounding box. All the four gures show two orders of structure functionsS2 and S8. The forced case is plotted in red, special case in green and decay case in blue from here-on in this entire section.
In the initial state, the decaying turbulence structure function has the highest magni-tude, the forced case comes second and the special case comes last, for bothS2 andS8. In the decaying case, this plot corresponds to the state of the system which for a short period of time has a large amount of energy before the actual decay process starts (see
5.2 Structure Functions and ESS 91 g. 4.2). For the forced case it is the state when turbulence has started (see g. 4.3).
Since the special case has its origin in the forced case, the plot here also corresponds to the case where turbulence has just kicked in. It is in fact at this point in time that the forcing is withdrawn from the system. The shape of the plots show a region over which all of them have a constant value. This trend changes in the nal stages of the simulation i.e. g. 5.2b, where the magnitude of the forced case dominates while the decaying case has the least magnitude of the three. These features in the forced case are a result of sustained input of magnetic helicity through the driving, while for the decaying case the energy is in dissipative phase, without any input. For the special case, it is tending towards the decaying case. The same trend is observed forb in g.
5.2 c and d.
An important property of structure functions is that they exhibit self-similar behavior, in the inertial range as Spc(`) = aplζp. Thus the knowledge of ap and ζp character-ize the statistical distribution of eddies in the inertial range [17, 20]. These scaling exponents are expected to be clearly visible in the logarithmic derivate plots of the structure functions as the derivatives asymptotically form a plateau at inertial-range scales. These plateaux appear in front of a fall-o of the curves at large scales. The log-arithmic derivatives approach a constant value immediately in front of this transition from inertial to large scales.
a)
Figure caption on page 93
b)
c)
Figure caption on page 93
5.2 Structure Functions and ESS 93
d)
Figure 5.2: Structure Functions ofS2andS8forz+ andb. a) initial states of all the three cases for z+, b) nal states of all the three cases forz+, c) initial states of all the three cases forband d) nal states of all the three cases forb. The three cases: red: forced, green: special and blue: decaying.
The initial times are t=1.28, t=0.36 and t=0.18 for forced, special and decaying cases respectively.
The nal times are t=6.66, t=5.89 and t=9.33 for forced, special and decaying cases respectively.
This value indicates the most probable scaling exponent of the structure functions at inertial range scales.
Logarithmic Derivatives
The logarithmic derivative of the structure functions are given by dlnSpc(`)/dln(l) and as already mentioned they show a atness in and around the inertial range. The y-axis value at which this atness occurs for the second order structure function, is indicated by ζ2 and it is related to the spectral power of the energy spectrum by α= 1 +ζ2, whereα is the magnitude of the power law [1]. Thus the y-axis component for this logarithmic derivative plot ofz+, at which atness occurs, serves as one of the conrmation methods for the spectral power law of energy, in turbulent systems. This plot becomes unstable as order increases because of the accumulation of statistical noise at high orders. The next set of plots in g. 5.3 a-d, show the corresponding logarithmic derivatives of the structure function plots of g. 5.2 a-d respectively. In the plots of g. 5.3 a and b, the logarithmic derivative for the structure functions of g. 5.2 a and b are shown.
a)
b)
Figure caption on page 95
5.2 Structure Functions and ESS 95
c)
d)
Figure 5.3: Logarithmic derivatives of structure functions ofS2andS8forz+andb. a) initial states of all the three cases forz+, b) nal states of all the three cases forz+, c) initial states of all the three cases forband d) nal states of all the three cases forb. The three cases: red: forced, green: special and blue: decaying.
The initial plots (g. 5.3a) do not yield the scaling function exponents as the turbulence in all the three cases is not fully developed. From the nal state the scaling exponents
are estimated. For the decay and special cases, a constant value is observed in the logarithmic derivative plot from 0.04 to 0.18for S2, which corresponds to the inertial range of k = 7−30 in the energy spectrum of the decay case (see g. 4.10c). The exponent value obtained is 0.8 with an error of ±0.05 in the initial part of the range (at 0.04 of the x-axis) to as high as ±0.2 (at 0.2 of the x-axis) in the nal part of the range, as seen from the horizontal line shown in the top plot of g. 5.3b. For the forced case, the plateau region starts at 0.15 and extends up to 0.6, but this does not correspond to any inertial range in the energy spectrum. The value of this plateau region is 1. However the plot joins the decay and special case curves briey between 0.07 to 0.09, which is the closest it gets near the inertial range. The actual expected exponent value is 0.66 but the associated tting procedure cause measurement errors (shown by the error bars in the plot) which are estimated by the vertical extension of the plateaux. The usage of hyperviscosity in the simulations may also have an eect on the structure formation process [66], probably aecting the determination of the value of the exponent. The higher order plot (S8) generally shows more irregular behavior owing to accumulation of statistical noise. Although the nature of the curve looks similar for magnetic eld, no such exponent estimates are made from its plots of g.
5.3 c and d as it is not an ideal invariant. It has also been observed that logarithmic derivative plots of lower order structure functions are more orderly than the higher order ones. Thus it is dicult to determine a structure function constantζp, for higher orders, as p becomes larger than 4 [17, 20]. The statistical noise levels are high for the higher order structure functions and errors are also high in this method. Thus an approach called extended self similarity (ESS) is used (see section 3.3.1), to get better understanding of the structure function exponents, principally at higher orders.
In ESS, all other structure functions are drawn relative to a lower order structure function, whose structure function exponent is unambiguously known, or known with minimum error.
ESS
The basic idea and equation for ESS was explained in the section 3.3.1 through the equations (3.39) and (3.40). They are reproduced here for further discussions.
Sp(Sr(`))∼(`ζr)ζp ∼`ξp,r, (5.5)
ζp =ξp,r/ζr. (5.6)
5.2 Structure Functions and ESS 97 Here ξp,r is the relative scaling exponent and equation (5.5) gives the prescription to attain the absolute scaling exponent ζp, when any other structure function Sp, is plotted relative toSr. In this work,Sr is the second order structure function, as it has a strong relation with the energy spectrum. Consistently, for magnetic eld too the second order structure function is used as the base.
a)
b)
Figure caption on page 98
c)
d)
Figure 5.4: Extended self similarity plots forS1 andS8 with respect to S2 forz+ andb. a) initial states of all the three cases forz+, b) nal states of all the three cases for z+, c) initial states of all the three cases forb and d) nal states of all the three cases for b. The three cases: red: forced, green: special and blue: decaying.
The advantages of this approach stem from the fact that structure functions of the same eld exhibit same kind of features in their shape. Thus when a structure function
5.3 Intermittency and Modeling 99